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From: olcott <polcott2@gmail.com>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
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Date: Thu, 28 Mar 2024 16:25:29 -0500
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On 3/28/2024 3:38 PM, Andy Walker wrote:
> On 28/03/2024 16:53, olcott wrote:
>>>> Yet it seems that wij is correct that 0.999... would seem to
>>>> be infinitesimally < 1.0.
>>>      That /cannot/ be correct in the "real" numbers, in which there
>>> are no infinitesimals [basic axiom of the reals].  In other systems of
>>> numbers, it could be correct, 
>> Yes.
>>> but that will depend on what is meant by
>>> "0.999..", 
>> Approaching yet never reaching 1.0.
> 
>      That is a property of the numbers 0.9, 0.99, 0.999 and so
> on arranged as a sequence [and of many other sequences], but is not
> /yet/ a value.  Not until you explain what you mean.  In conventional
> mathematics, it is usually taken to mean the limit of that sequence
> expressed as a real number, where "limit" has a precise meaning as
> discussed and formalised in the 19thC.  That limit is 1.  

I disagree yet prior to my infinitesimal number system there was no
way to say this. 0.999... is exactly one geometric point less than 1.0
the same way that this line segment [0.0,1.0] > [0.0,1.0) by exactly
one geometric on the number line.

> Not a tiny
> bit less than one, not some new sort of object, but 1, exactly.  You
> and Wij may find that surprising, or even nonsensical, but it is what
> the mathematics tells us from the axioms of the real numbers and from
> the definition of "limit".  If you want the answer to be different,
> then that must follow from different axioms and definitions.  Until
> you and/or Wij tell us what those are, there is nothing further useful
> to be said.
> 

Yes we all agree that 0.999... never gets to 1.0.

>>> and note that if you appeal to something that mentions limits
>>> to define this, then you have to explain how infinite and infinitesimal
>>> numbers are handled in the definition.
> 
>      Again, there are no infinite or infinitesimal real numbers, so
> if you want an infinitesimal in your answer, it is incumbent on you to
> explain what you are using /other than/ conventional maths.
> 

Been there done that.

>>>>                   One geometric point on the number line.
>>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>>      Until you describe the axioms of what you mean by "geometric
>>> point" and "number line", this is meaningless verbiage.  Give your
>> Of course by geometric point I must mean a box of chocolates and by
>> number line I mean a pretty pink bow. No one would ever suspect that
>> these terms have their conventional meanings.
> 
>      I didn't ask what "geometric point" and "number line" are, but
> what axioms you think they have.  In conventional mathematics, those two
> intervals have /exactly/ the same measure even though they are not
> exactly the same sets of points.  

That is inconsistent. They are exactly the same points up until
the last point at 1.0 is reached by one yet not the other.

> If you get a different answer [and
> have not simply made a mistake], it /must/ be because you are using
> different axioms.  What are they?
> 

I am just showing EXACTLY where the conventional notions lead.

>>> axioms, and it becomes possible to discuss this.  Until then, we are
>>> entitled to assume that you and Wij are talking about the "traditional"
>>> "real" numbers [as used in engineering, etc.] in which there are no
>>> infinitesimals, and so the only interpretation we can make of the size
>>> of "one geometric point" is the usual "measure", which is zero.
>> Yet it is never actually zero because it is possible to specify a
>> line segment that is exactly one geometric point longer than another.
>> [0.0, 1.0] - [0.0, 1.0) = one geometric point.
> 
>      But "one geometric point" has measure zero.  Not "never actually

I just proved otherwise. [0.0, 1.0] has all of the same points
as [0.0, 1.0) except that it has one more point.

> zero", but actually and really zero.  Unless, that is, you are using some
> different and as yet unexplained axioms/definitions.  Which are ...?
> 

Conventional interval notion proves otherwise.
Most logicians abhor thinking outside-the-box.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer